The peak error in the controlled variable for load upsets is proportional to the ratio of the total loop dead time to the open-loop process time constant:
To estimate the savings from elimination of excess reagent, reactant or reflux flow, the amplitude from Equations 1 or 3 or the peak error from Equation 8 is divided by the product of the gains of the process variable and controlled variable to get back to the change in flow:
For back mixed volumes, the amplitude of the oscillation in the reagent flow should be multiplied by process gain and an amplitude ratio from frequency response that is proportional to loop oscillation period and inversely proportional to the mixing time constant to get a filtered amplitude of oscillations in a key process variable of interest:
The process gains in Equations 9 and 10 are different for different process variables or for processes with operating point nonlinearities (e.g., pH). The elimination of the oscillation in the process variable can translate to a proportional shift to a more optimum set point or a reduction in scrap or downgraded product.
Solution in the Selection
Control valves designed for on-off operation generally are a poor choice for a control valve. Conversely, throttling valves should not be used for isolation or interlocks.
A calculation can be made to compensate for backlash by adding a bias to the controller output whenever a signal reversal exceeds a noise band. However, the exact dead band is a moving target and a bias greater than the actual dead band effectively creates slip. Also, stick-slip goes hand-in-hand with dead band in high-performance valves.
The control valve that responds best to small changes is a judiciously sized sliding stem (globe) valve with a digital positioner and a correctly sized diaphragm actuator and properly tightened Teflon packing. It has negligible backlash and a stick-slip of 0.1%. If you must use a rotary valve, avoid tight shutoff and high-friction packing and use a diaphragm actuator with a short shaft and splined connections between the actuator shaft and the ball, disc or plug stem. Make sure the ball, disc or plug are cast with its stem otherwise the junction between the ball, disc or plug and its stem can become another source of backlash. If high temperature or environmental packing must be used, increase the actuator size and positioner gain to help it better deal with the packing's increased friction.
Big Problems for Small Valves
Small valves are more prone to improper sizing, irregular flow characteristics, greater stick-slip and plugging. Here, size does matter because most of these problems originate from extremely small Reynolds Numbers, clearances, and stem diameters.
For a CV less than 0.01 or extremely viscous fluids, the valve may be operating in the laminar flow regime or in the region of transition from turbulent to laminar flow. Liquid flow moves from being approximately proportional to the square root of the pressure drop towards being proportional to the pressure drop as the flow goes from being fully turbulent to completely laminar. For anything other than a 1-psi pressure drop this can translate to an enormous sizing error. The result is often an oversized valve that rides the seat, where the high seating friction causes excessive stick-slip. Even worse, operation in the transition region where the flow for a particular valve position has poor reproducibility, because normally insignificant disturbances such as microscopic changes in roughness or small vibrations can trigger a switch between turbulent and laminar flow and an erratic installed flow characteristic.
If you also consider the possibility of a significant distortion of the inherent flow characteristic caused by machining tolerances that are an appreciable portion of the clearances for such tiny trim sizes, the scene is set up for an unknown and extreme nonlinearity. Tiny clearances can pose all sorts of problems because small particles and coatings cause plugging and sticking. The low flow velocities at the surface that is normally associated with laminar flow makes the likely hood of coating much greater.
Finally, tiny stems are likely to be bent from normal handling both before and after installation. Slight deflections of the stem can cause huge amounts of stick-slip. What good is a control valve if you can't drop it or step on it?
Is the situation hopeless? Not if you go with a manufacturer who specializes in tighter machining tolerances and minimizing stick-slip. I would stay away from stems smaller than 3/8 inch and I would insist on getting response and flow test results. I would use computer programs now available that properly deal with laminar flow and offer an installed characteristic curve for your piping and operating conditions. Even though a smart digital positioner may be larger and cost more than the valve, it is more important than ever that it be used and properly tuned for the small actuator volume. The challenge here is for packaged platforms that are going for low bid not to get cheap where it hurts.
If there is a tendency for plugging, pulse-width modulation is a solution if there is sufficient back-missed volumes to attenuate the pulses. This also can provide a linear flow characteristic and a flow large enough to be turbulent. The ratio of maximum to minimum pulse width establishes the rangeability. The maximum pulse width, and hence, cycle time determine the degree of variability that needs to be filtered and the additional dead time from the pulse off time. Now stroking time can be an issue because it is desirable to have the minimum pulse be as short as possible.
A variable-speed drive is another possible solution, but the user needs to be aware of a dead band that is artificially introduced into the electronics and the minimum discharge head requirements to prevent reverse flow for varying static heads.
Interestingly enough, valve specifications do not require that a control valve move. A response requirement should be added to the control valve specification that details the stick-slip, dead band, and the response time (63) for a small step in the throttle range. Ideally, a ramp at the expected rate of change of the loop should be used rather than a step, to reveal the hidden dead time from dead band and the saw tooth from stick-slip.